Your math teacher may answer like this: “There is no biggest number. Because, if such a number existed, you could add 1 to it to create an even bigger number, so the original number would not be the biggest anymore.”

This answer is perfectly valid, using a proof called “proof by contradiction.” It simply tells us there is no biggest number. This conclusion may not be so satisfying, so let’s stick to our question a bit more.

What I did when I was a kid was create a hypothetical “biggest number,” and imagine what kind of properties it would have. A child version of me called this number "T" (an initial of some cool-sounding Japanese word I made up). T must always be the biggest, strongest, most superior number among all numbers. How would T behave?

If you add/subtract 1 (or any finite number) to/from T, it shouldn’t affect T at all, because 1 is so insignificant compared to T. So I wrote down in my notebook, proudly, crazy equations like T + 1 = T, T – 1 = T, etc. These equations would be mathematically wrong if T were an ordinary number.

What if we add another T to T itself? In other words, what if T is doubled? My promise was to let T be the biggest number no matter what. T can’t surrender the throne to any other number, even if that was its own double. So, I wrote down another crazy equation T + T = T (namely, 2 × T = T). Exploring new properties of a new number of my own creation this way was a lot of fun for the young me.

Then I learned in high school that there is indeed a concept called “infinity” in math, symbolized as “∞,” which had the exact same properties as I had dreamed of. So I wasn’t crazy after all! But infinity is not a number in a traditional sense, so mathematicians don’t use it in equations like “∞ + 1 = ∞.” A more careful handling is necessary for infinity.

Moreover, much later on in college, I also learned that there is more than one type of infinity. There are infinities whose “sizes” are bigger than other infinities! The competition among different infinities goes on and on, with no apparent limit. Understanding this would require more advanced mathematical knowledge, so I will skip it for now.

A take-home message is that it is perfectly okay to break the rules of traditional math and create a new number or concept and study its properties. After all, that is how math has developed. There were times when there was no zero, no negative number, no infinity. People who were bold enough to break the conventions at their times were the ones who brought innovations and breakthroughs in math (and also in science, engineering and technology). You can be one of them, too.

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